My interest in this subject extends from daily experiences with
geometry and typography over a 35-year career of visual design. The
terminology is personal and was created as needed.

5 x 5 ?

In late 1986, while sketching on a quadrille pad, I generated this
little drawing and asked myself a seemingly simple question:

How many different 5x5 images will nature allow?

After filling pages of quadrille pad I realized that imagination alone
wasn't up to the task (not an easy thing for an artist to accept). I next
made a 20-foot-wide wallchart and kept searching for a method to generate
quantities of these symbols.

Invalid Ordering

Over several years I worked on the 5x5 problem in my spare time. It
soon became my favorite intellectual diversion. I thought of it as some
kind of hyper-digital I Ching. I thought of it as my Glass Bead
Game.

I identified all the expressions possible on a five-unit horizontal
line. The first image to the right shows the initial ordering system.

Though an invalid approach, this ordering system produced some
interesting results, such as the middle image on the right, which shows
repetitive patterning in widths 1, 3, 5, and 7.

The final image on the right, titled Odd Growth, shows
remarkable pattern repetition. It is composed of the symmetric elements
for (odd) widths 1 through 13.

This image was the first to display what I came to call
Completion. Imagine folding the image at the center (a near ratio
actually); the positive image above completely covers the negative below.
This concept of Completion grew as the project progressed.

These stacked-element images are called Signatures.

Valid Ordering

The ordering method I had been using did not produce the beautiful
results hoped for. The method was problematic rather than problem
solving.

Finally, in 1996 I had an insight. Everything fell together and I was
able to produce most of the images shown here in just a weekend. The
insight of course was the relationship between my ordering problem and
binary numbers. (Some newly acquired computer skills helped.)

Because there are so many possible combinations of the 32 elements, I
decided to look at the limited set of symmetric elements shown in red on
the right.

This limitation allowed me to inspect 16,807 examples in the 7
Generation charts (see further on), where each 5x5 symbol is constructed
entirely from symmetric elements.

This red symbol is termed an Icon.

Text

Utilizing the seven elements of the Icon above, I methodically
built these 2x5 units and called them Text.

I was struck by the reflective symmetry within this chart, delineated
by the descending diagonal indicated in red.

Implied Subtext

Implied

I noticed a relationship between the 2x5 units and their 3x5
counterparts; I termed this relationship Implied.

This Subtext set is the 3x5 implied counterpart of the Text set
shown above. However, there are 7 (natural) Subtext sets. The first
horizontal grouping in the chart at right is the first row of the Subtext
1 chart (shown further on); the second row is the second row of the
Subtext 2 chart, and so on.

Implied Primes

Placing 2x5 Text over 3x5 Subtext became the Method for
generating large numbers of symbols.

When the Text and Subtext components that have an implied relationship
are combined, the resulting images are Primes. The radial Primes
are shown in red.

Subtext

There is no reflection in the following seven Subtext charts, which are
a repetition of the Text set but with one of the symmetric elements in the
top position. This top line of the Subtext is the ruling line of the
symbol and lies at the center of Text/Subtext composites. This concept led
to a deeper understanding of a center line (division) and two fields or
sets. This is expressed both in the individual symbols and in the diagonal
reflection evident in the Generation charts.

Subtext 1

Subtext 2

Subtext 3

Subtext 4

Subtext 5

Subtext 6

Subtext 7

Primes

Primes are the spine (descending diagonal) of all the Generation charts
(shown further on). These sets contain 90º rotation examples as well as
radial singularities, which are indicated in red.

Primes 1

Primes 2

Primes 3

Primes 4

Primes 5

Primes 6

Primes 7

Generation

Working in Photoshop, in bitmap mode and 72dpi, I placed all the text
elements across the top of the chart, and one of the seven Subtext sets
down the side for each chart generated. Selecting, copying,
step-and-repeat from the top down and left to right constructs the full
range of possibilities. Once again the reflection across the diagonal is
plainly evident.

Generation 1

Generation 2

Generation 3

Generation 4

Generation 5

Generation 6

Generation 7

Imagination

The images above are but a snapshot of far larger (imaginary) charts
composed of all 31 visible and one invisible, for a total of 32 elements.
Such charts would be nearly 8 feet on a side at 72dpi in one-point
resolution (given that each symbol naturally rests upon a 7x7 tile and
given that each chart will be 1024x1024 symbols plus one more for
generation is 7x1025=7175 pixels). Since I don't have acces to a printer
that size, and because one-point resolution (the charts above are 2-point
resolution) would require a photo loop to read, I haven't produced those
32 charts. It would be interesting, however, to color code the various
categories in such charts and observe the resulting patterns. (6.14Mb
bitmap, 147.3 Mb rgb, 196.4 Mb cmyk)

Classification

The generation charts shown above were originally produced in bitmap
mode. Once the variety was revealed, I arrived at the simple
classification shown at right.

1 too massive,2 checkerboard3 floaters4 one-stroke,
letter-like

(Of course there are combinations of two or more categories that I
ignored, but might look into later.)

I colored the charts above, removed all the gray, slid the black
together, and made the gleaned charts below to reveal those category 4
sets that had inspired me in the first place.

Gleans

The original motivation for this project was the promise of inspecting
all the available one-stroke, letter-like symbols. The visual power of
these reduced charts arises from both the symbolic nature of the images
and the stark presentation of reflection that they portray. On some of
these charts I included floaters so as to fill in and make the chart
larger and more interesting.

Glean 1

Glean 2

Glean 3

Glean 4

Glean 5

Glean 6

Glean 7

Completion

It is clear that almost 98% of 5x5 symmetry is composed of (180º)
reflective pairs like the black symbols in the Glean 7 chart above.

What remains are the 343 Primes that define the descending red
diagonals. Most are (90º) rotational pairs; those symbols that possess
full rotational symmetry are pairless.

The idea of Completion led to a recognition of the special nature of
these (pairless) singularities.

Completion is what I term the pairing of the positive and negative
image for any given pattern.

This chart shows 32 radial Completion pairs and is organized to reveal
the structural relationships among the pairs of pairs it displays.

512 Permutations of 3 x 3

Recently, I took what I had learned from my study of 5x5 and
applied it to 3x3.

At 512 permutations, this 3x3 project had a more discreet path and
required comparatively little time. In addition, this exercise allowed me
to look at both the symmetric and asymmetric elements and is small enough
to include the line of all zeros in its construction.

The diagonal reflection explained above is evident in these charts.

But something more is clear. The miniscule potential of 3x3 gives
little notice of the veritable explosion of potential at 5x5. This
suggests evidence of an irreducible complexity at 5x5.

The relationship between the patterns presented here and the digital
patterns that have become, over the previous two decades, the defining
paradigm of our age is also worth noting.

3 x 3 Completion Pairs

The concept of Completion is fully tested in the following chart. With
all 512 permutations of 3x3, 256 Completion pairs resulted. Each pair is
assembled beginning at the front of the first chart (above) and the end of
the last chart, and peals off in sequence as intuited.

An Interesting Question

As I was finishing up the 3x3 charts, I created the image at right. It
is composed of the 3x3 radial Completion pairs, which are arranged to
reflect across the chart. The question that nags me is, what (if anything)
occupies that center square?

Perception

Both the Christian cross and the swastika are extant in the
permutations of 5x5. Moreover, many symbols of ancient origin and meaning
seem evident as well. There appears to be a very fundamental pattern
recognition tropism at work. And yet, when put to close examination, few
specific examples can be found in any cultural context.

This vague familiarity and lack of specificity point to a fundamental,
perhaps hard-wired, pattern recognition.

As part of nature's allowance, we humans are likely subject to the same
compelling efficiencies as our host. Human ideation seems to resonate in
some primitive way with these simple (yet complex) potentials.

This resonance may in part explain my compulsion to create, inspect,
and play with these images for the past two decades.

Mark Hess

Mark Hess is a self-employed visual designer now living in Trinidad,
Colorado after 30 years in Denver. He can be contacted at: